Supposing that it is 9 to 7 against a person A who is now 35 years of age living till he is 65, and 3 to 2 against a person B now 45 till he is 75, find the chance that one at least of these persons will be alive 30 years hence.

Answer

To solve the problem step by step, we will first determine the probabilities of both individuals A and B being alive after 30 years, and then use these probabilities to find the chance that at least one of them will be alive.

### Step 1: Determine the probability of person A being alive in 30 years
- The odds against person A living until 65 are given as 9 to 7. This means:
\[ \text{Odds against} = \frac{P(A \text{ not alive})}{P(A \text{ alive})} = \frac{9}{7} \]
- Let \( P(A \text{ alive}) = p_A \) and \( P(A \text{ not alive}) = 1 - p_A \).
- From the odds, we have:
\[ \frac{1 - p_A}{p_A} = \frac{9}{7} \]
- Cross-multiplying gives:
\[ 7(1 - p_A) = 9p_A \]
\[ 7 - 7p_A = 9p_A \]
\[ 7 = 16p_A \]
\[ p_A = \frac{7}{16} \]

### Step 2: Determine the probability of person B being alive in 30 years
- The odds against person B living until 75 are given as 3 to 2. This means:
\[ \text{Odds against} = \frac{P(B \text{ not alive})}{P(B \text{ alive})} = \frac{3}{2} \]
- Let \( P(B \text{ alive}) = p_B \) and \( P(B \text{ not alive}) = 1 - p_B \).
- From the odds, we have:
\[ \frac{1 - p_B}{p_B} = \frac{3}{2} \]
- Cross-multiplying gives:
\[ 2(1 - p_B) = 3p_B \]
\[ 2 - 2p_B = 3p_B \]
\[ 2 = 5p_B \]
\[ p_B = \frac{2}{5} \]

### Step 3: Calculate the probability that at least one of them is alive
- We need to find the probability that at least one of them is alive, which can be calculated using the complement:
\[ P(\text{at least one alive}) = 1 - P(A \text{ not alive}) \cdot P(B \text{ not alive}) \]
- We already have:
\[ P(A \text{ not alive}) = 1 - p_A = 1 - \frac{7}{16} = \frac{9}{16} \]
\[ P(B \text{ not alive}) = 1 - p_B = 1 - \frac{2}{5} = \frac{3}{5} \]
- Now we can compute:
\[ P(A \text{ not alive}) \cdot P(B \text{ not alive}) = \frac{9}{16} \cdot \frac{3}{5} = \frac{27}{80} \]
- Therefore:
\[ P(\text{at least one alive}) = 1 - \frac{27}{80} = \frac{80 - 27}{80} = \frac{53}{80} \]

### Final Answer
The probability that at least one of the persons A or B will be alive in 30 years is:
\[ \frac{53}{80} \]